Paul H. Ginsparg

The structure of gauge and gravitational anomalies

It is shown how the form of the gauge and gravitational anomalies in quantum field theories may be derived from classical index theorems. The gravitational anomaly in both Einstein and Lorentz form is considered and their equivalence is exhibited. The formalism of gauge and gravitational theories is reviewed using the language of differential geometry, and notions from the theory of characteristic classes necessary for understanding the classical index theorems are introduced. The treatment of known topological results includes a pedagogical derivation of the Wess-Zumino effective Lagrangian in arbitrary even dimension. The relation between various forms of the anomaly present in the literature is also clarified.

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2n dimensions is related to the non-trivial topology of gauge orbit space. The form of the anomaly and its normalization are shown to follow from a familiar index theorem for a certain (2n + 2)-dimensional Dirac operator. We are thus able to recover and give topological meaning to a variety of results concerning anomalies in 4- and higher-dimensional theories.

Curiosities at c = 1

We consider conformal field theories on a torus with central charge c = 1, and in particular models based upon modding out string propagation on the SU(2) group manifold by its finite subgroups. We find that the partition functions for these models coincide with the continuum limit partition functions of a recently introduced class of RSOS models, defined in terms of the extended Dynkin diagrams of simply-laced Lie algebras, thus giving an alternative interpretation for the primary fields in these latter theories. Three of the models have no massless moduli and thus do not lie on the same line of critical points with the rest. The particular correspondence between simply-laced Lie algebras and finite subgroups of SU(2) that emerges coincides with that which has already appeared in other mathematical contexts.

2D gravity+1D matter

Abstract We consider a formulation of nonperturbative two-dimensional quantum gravity coupled to a single bosonic field (d = 1 matter). Starting from a matrix realization of the discretized model, we express the continuum theory as a double scaling limit in which the 2D cosmological constant g tends towards a critical value gc, and the string coupling 1 N → 0, with the scaling parameter α∝1n(g-gc)(g-gc)N held fixed. We find that in this formulation logarithmic corrections already present at tree level persist to all higher genus, suggesting a behavior different from the previously considered cases of d

Semiclassical perdurance of de Sitter space

We investigate the classical and semiclassical instabilities of de Sitter space. Due to the presence of a cosmological event horizon and its associated Hawking temperature, de Sitter space might be expected to behave in some respects like flat space at finite temperature. We use the euclidean formulation of quantum gravity to show that de Sitter space does exhibit a semiclassical instability to the nucleation of black holes. We find, however, no analog to the classical instability due to gravitational clumping of thermally excited gravitons.

An O(16) × O(16) heterotic string

By modifying the definition of the fermion number projection in the E 8 × E 8 heterotic superstring theory in a manner consistent with modular invariance and factorization, a tachyon-free string theory is obtained without spacetime supersymmetry. The low energy limit is a ten-dimensional anomaly-free chiral SO(16) × SO(16) gauge theory coupled to gravity.

Toroidal compactification of non-supersymmetric heterotic strings

We show that all the recently discovered rank-16 non-supersymmetric heterotic strings form a single connected class of theories when compactified on tori in the presence of constant background fields. In addition we show that in a weaker sense all the heterotic theories compactified on tori are connected. We also discuss the dependence of the higher loop amplitudessb in the heterotic string on the background fields, and show that whenever the compactified theory has an enhanced gauge symmetry without U(1) factors, the cosmological constant to all loops is an extremum with respect to the background field expectation values.

Gauge and gravitational couplings in four-dimensional string theories

Abstract We reconsider the relation between the string scale m s = 2 α′ and the Planck scale m Pl = 1 G N in lower-dimensional string theories that do not necessarily admit an interpretation as compactified higher-dimensional theories. Assuming that the spacetime gauge symmetry is the reflection of a non-abelian affine Kac-Moody algebra on the world sheet, we point out that the string scale is generally given by m s 2 m Pl 2 =( g 2 4π )k , where g is the spacetime gauge coupling and k is the integrally quantized level of the affine algebra.

Finiteness of Ricci Flat Supersymmetric Nonlinear Sigma Models

Combining the constraints of Kähler differential geometry with the universality of the normal coordinate expansion in the background field method, we study the ultraviolet behavior of 2-dimensional supersymmetric non-linear σ-models with target space an arbitrary riemannian manifoldM. We show that the constraint ofN=2 supersymmetry requires that all counterterms to the metric beyond one-loop order be cohomologically trivial. It follows that such supersymmetric non-linear σ-models defined on locally symmetric spaces are super-renormalizable and thatN=4 models are on-shell ultraviolet finite to all orders of perturbation theory.

The Ising Model, the Yang-Lee Edge Singularity, and 2D Quantum Gravity

Abstract We consider one-and two-matrix models recently used to define nonperturbative 2D quantum gravity. We prove the equivalence of the m = 3 multicritical one-matrix model and the Yang-Lee edge singularity coupled to 2D quantum gravity. We also derive formulae for the correlation functions for the Yang-Lee edge singularity and for the Ising model in their summed over topologies formulations.